Related papers: Pinned Dot Product Set Estimates

We study an extension of the Falconer distance problem in the multiparameter setting. Given $\ell\geq 1$ and $\mathbb{R}^{d}=\mathbb{R}^{d_1}\times\cdots \times\mathbb{R}^{d_\ell}$, $d_i\geq 2$. For any compact set $E\subset \mathbb{R}^{d}$…

We introduce and study the \emph{Fourier spectrum} which is a continuously parametrised family of dimensions living between the Fourier dimension and the Hausdorff dimension for both sets and measures. We establish some fundamental theory…

Let $S \subset {\mathbb R}^d$ be contained in the unit ball. Let $\Delta(S)=\{||a-b||:a,b \in S\}$, the Euclidean distance set of $S$. Falconer conjectured that the $\Delta(S)$ has positive Lebesque measure if the Hausdorff dimension of $S$…

We consider several distances between two sets of points, which are modifications of the Hausdorff metric, and apply them to describe some fractals such as $\delta$-quasi-self-similar sets, and some other geometric notions in Euclidean…

Self-similar sets require a separation condition to admit a nice mathematical structure. The classical open set condition (OSC) is difficult to verify. Zerner proved that there is a positive and finite Hausdorff measure for a weaker…

We compute the Hausdorff dimension of sets defined by the growth of weighted products of multiple digits at arbitrary positions in $d$-decaying Gauss-like iterated function systems. We provide the complete Hausdorff dimensional result for…

We show that if compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension larger than $\frac{d}{2}+\frac{1}{4}$, where $d\geq 4$ is an even integer, then the distance set of $E$ has positive Lebesgue measure. This improves the previously…

We determine the Hausdorff, packing and box-counting dimension of a family of self-affine sets generalizing Bara\'nski carpets. More specifically, we fix a Bara\'nski system and allow both vertical and horizontal random translations, while…

We study the scaling scenery of Gibbs measures for subshifts of finite type on self-conformal fractals and applications to Falconer's distance set problem and dimensions of projections. Our analysis includes hyperbolic Julia sets, limit…

We consider limit sets of some conformal iterated function systems, and introduce classes of subsets of the limit set, with the property that the classes are closed under countable intersections and all sets in the classes have large…

Fundamental questions in Diophantine approximation are related to the Hausdorff dimension of sets of the form $\{x\in \mathbb{R}: \delta_x = \delta\}$, where $\delta \geq 1$ and $\delta_x$ is the Diophantine approximation rate of an…

We study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several…

We prove bounds for the almost sure value of the Hausdorff dimension of the limsup set of a sequence of balls in $\mathbf{R}^d$ whose centres are independent, identically distributed random variables. The formulas obtained involve the rate…

In this paper, the product of the Hausdorff metric on the product space is defined and the equivalency between the product Hausdorff metric and the Hausdorff metric on the product space is established. The finite product of the iterated…

We study dimensional properties of visible parts of fractal percolation in the plane. Provided that the dimension of the fractal percolation is at least 1, we show that, conditioned on non-extinction, almost surely all visible parts from…

The Hausdorff distance is a measure of (dis-)similarity between two sets which is widely used in various applications. Most of the applied literature is devoted to the computation for sets consisting of a finite number of points. This has…

We say that $E$ is a microset of the compact set $K\subset \mathbb{R}^d$ if there exist sequences $\lambda_n\geq 1$ and $u_n\in \mathbb{R}^d$ such that $(\lambda_n K + u_n ) \cap [0,1]^d$ converges to $E$ in the Hausdorff metric, and…

In this paper, we construct new multifractal measures, on the Euclidean space $\mathbb{R}^n$, in a similar manner to Hewitt-Stomberg measures but using the class of all $n$-dimensional half-open binary cubes of covering sets in the…

For $P$ a poset, the dimension of $P$ is defined to be the least cardinal $\kappa$ such that $P$ is embeddable in a direct product of $\kappa$ totally ordered sets. We study the behavior of this function on finite-dimensional (not…

Hausdorff dimension of level sets of generic continuous functions defined on fractals can give information about the "thickness/narrow cross-sections" "network" corresponding to a fractal set, $F$. This lead to the definition of the…